If `(x-(1)/(x))=4`, then the value of `(x^(2)+(1)/(x^(2)))` is :

Options

A. 16
B. 18
C. 14
D. none of these

Answer

To solve the equation \( x - \frac{1}{x} = 4 \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x - \frac{1}{x} = 4 \]

### Step 2: Square both sides
Squaring both sides of the equation gives us:
\[ \left( x - \frac{1}{x} \right)^2 = 4^2 \]
This simplifies to:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 16 \]
which further simplifies to:
\[ x^2 - 2 + \frac{1}{x^2} = 16 \]

### Step 3: Rearranging the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} - 2 = 16 \]
Adding 2 to both sides gives:
\[ x^2 + \frac{1}{x^2} = 16 + 2 \]

### Step 4: Final calculation
Thus, we find:
\[ x^2 + \frac{1}{x^2} = 18 \]

### Conclusion
The value of \( x^2 + \frac{1}{x^2} \) is \( 18 \).

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